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32,000 | Given an isosceles triangle with a vertex angle of 36°, the ratio of the base to the leg is equal to . | \frac{\sqrt{5}-1}{2} | 69.53125 |
32,001 | Given that 2 teachers and 4 students are to be divided into 2 groups, each consisting of 1 teacher and 2 students, calculate the total number of different arrangements for the social practice activities in two different locations, A and B. | 12 | 4.6875 |
32,002 | Determine the slope \(m\) of the asymptotes for the hyperbola given by the equation
\[
\frac{y^2}{16} - \frac{x^2}{9} = 1.
\] | \frac{4}{3} | 5.46875 |
32,003 | Given that the sum of the first $n$ terms of the sequence ${a_n}$ is $S_n=\frac{n^2+n}{2}+1$, find the sum of the first 99 terms of the sequence ${\frac{1}{a_n a_{n+1}}}$, denoted as $T_{99}$. | \frac{37}{50} | 66.40625 |
32,004 | An ancient civilization has a tribe of 12 members organized hierarchically. The tribe has one main chief, two supporting chiefs (Senior and Junior), and each supporting chief has three inferior officers. If the tribe has 12 members in total, in how many ways can the leadership structure of the tribe be formed under these restrictions? | 2217600 | 11.71875 |
32,005 | The function $\mathbf{y}=f(x)$ satisfies the following conditions:
a) $f(4)=2$;
b) $f(n+1)=\frac{1}{f(0)+f(1)}+\frac{1}{f(1)+f(2)}+\ldots+\frac{1}{f(n)+f(n+1)}, n \geq 0$.
Find the value of $f(2022)$. | \sqrt{2022} | 0 |
32,006 | The diagram shows three rectangles and three straight lines. What is the value of \( p + q + r \)?
A) 135
B) 180
C) 210
D) 225
E) 270 | 180 | 56.25 |
32,007 | Determine the number of digits in the value of $2^{15} \times 5^{11}$. | 12 | 0.78125 |
32,008 | The sum of the largest number and the smallest number of a triple of positive integers $(x,y,z)$ is the power of the triple. Compute the sum of powers of all triples $(x,y,z)$ where $x,y,z \leq 9$. | 7290 | 0 |
32,009 | Given that $\overrightarrow{a}$ and $\overrightarrow{b}$ are unit vectors, if $|\overrightarrow{a}-2\overrightarrow{b}|=\sqrt{3}$, then the angle between $\overrightarrow{a}$ and $\overrightarrow{b}$ is ____. | \frac{1}{3}\pi | 0 |
32,010 | What is the sum of the six positive integer factors of 30, each multiplied by 2? | 144 | 12.5 |
32,011 | Given that an ellipse and a hyperbola share common foci $F_1$, $F_2$, and $P$ is one of their intersection points, $\angle F_1PF_2=60^\circ$, find the minimum value of $e_1^2+e_2^2$. | 1+\frac{\sqrt{3}}{2} | 9.375 |
32,012 | What is the result when $1\frac{1}{3}$ is subtracted from $2\frac{5}{6}$? | \frac{3}{2} | 20.3125 |
32,013 | Given a circle described by the equation $x^{2}-2x+y^{2}-2y+1=0$, find the cosine value of the angle between the two tangent lines drawn from an external point $P(3,2)$. | \frac{3}{5} | 40.625 |
32,014 | How many different rectangles with sides parallel to the grid can be formed by connecting four of the dots in a $5\times 5$ square array of dots? | 100 | 11.71875 |
32,015 | Calculate $1010101_2 + 111000_2$ and write your answer in base $10$. | 141 | 39.84375 |
32,016 | The equation of the asymptotes of the hyperbola \\(x^{2}- \frac {y^{2}}{2}=1\\) is \_\_\_\_\_\_; the eccentricity equals \_\_\_\_\_\_. | \sqrt {3} | 0 |
32,017 | The area of a triangle \(ABC\) is \(\displaystyle 40 \text{ cm}^2\). Points \(D, E\) and \(F\) are on sides \(AB, BC\) and \(CA\) respectively. If \(AD = 3 \text{ cm}, DB = 5 \text{ cm}\), and the area of triangle \(ABE\) is equal to the area of quadrilateral \(DBEF\), find the area of triangle \(AEC\) in \(\text{cm}^2\). | 15 | 13.28125 |
32,018 | In rectangle $PQRS$, $PQ=7$ and $QR =4$. Points $X$ and $Y$ are on $\overline{RS}$ so that $RX = 2$ and $SY=3$. Lines $PX$ and $QY$ intersect at $Z$. Find the area of $\triangle PQZ$. | 19.6 | 0 |
32,019 | Each of 8 balls is randomly and independently painted either black or white with equal probability. Calculate the probability that every ball is different in color from more than half of the other 7 balls. | \frac{35}{128} | 37.5 |
32,020 | Given that $a$, $b$, and $c$ are the sides opposite to angles $A$, $B$, and $C$ respectively in $\triangle ABC$, and $\sin A$, $\sin B$, and $\sin C$ form a geometric sequence. When $B$ takes the maximum value, the maximum value of $\sin A + \sin C$ is _____. | \sqrt{3} | 67.96875 |
32,021 | In a sequence $a_1, a_2, . . . , a_{1000}$ consisting of $1000$ distinct numbers a pair $(a_i, a_j )$ with $i < j$ is called *ascending* if $a_i < a_j$ and *descending* if $a_i > a_j$ . Determine the largest positive integer $k$ with the property that every sequence of $1000$ distinct numbers has at least $k$ non-overlapping ascending pairs or at least $k$ non-overlapping descending pairs. | 333 | 0 |
32,022 | Let $q(x)$ be a quadratic polynomial such that $[q(x)]^2 - x^2$ is divisible by $(x - 2)(x + 2)(x - 5)$. Find $q(10)$. | \frac{110}{7} | 7.8125 |
32,023 | Let \( n, m \) be positive integers. Define the sets \( A = \{1, 2, \ldots, n\} \) and \( B_{n}^{m} = \{ (a_1, a_2, \ldots, a_m) \mid a_i \in A, i = 1, 2, \ldots, m \} \) that satisfy:
1. \( |a_i - a_{i+1}| \neq n-1 \) for \( i = 1, 2, \ldots, m-1 \);
2. Among \( a_1, a_2, \ldots, a_m \) (with \( m \geq 3 \)), there are at least three different elements.
Find the number of elements in \( B_{n}^{m} \) and \( B_{6}^{3} \). | 104 | 62.5 |
32,024 | In triangle $ABC$, where $AB = \sqrt{34}$ and $AC = 5$, the angle $B$ is $90^\circ$. Calculate $\tan A$. | \frac{3}{5} | 67.1875 |
32,025 | Given the function $f(x)=\sin^{2}x+\sqrt{3}\sin x\sin (x+\frac{\pi}{2})$.
1. Find the value of $f(\frac{\pi}{12})$;
2. Find the maximum and minimum values of the function $f(x)$ when $x\in[0,\frac{\pi}{2}]$. | \frac{3}{2} | 0 |
32,026 | There is a unique two-digit positive integer $t$ for which the last two digits of $13 \cdot t$ are $26$. | 62 | 0 |
32,027 | Given that the function $f(x)$ satisfies $f(x+y)=f(x)+f(y)$ for all real numbers $x, y \in \mathbb{R}$, and $f(x) < 0$ when $x > 0$, and $f(3)=-2$.
1. Determine the parity (odd or even) of the function.
2. Determine the monotonicity of the function on $\mathbb{R}$.
3. Find the maximum and minimum values of $f(x)$ on $[-12,12]$. | -8 | 82.8125 |
32,028 | As shown in the diagram, four small plates \( A, B, C, D \) are arranged in a circular shape, with an unspecified number of candies placed on each plate. In each move, it is allowed to take all candies from 1, 3, or 4 plates, or from 2 adjacent plates. What is the maximum number of different possible amounts of candies that can be taken out? Please provide a reasoning. | 13 | 24.21875 |
32,029 | If the real numbers x and y satisfy \((x-3)^{2}+4(y-1)^{2}=4\), find the maximum and minimum values of \(\frac{x+y-3}{x-y+1}\). | -1 | 10.9375 |
32,030 | Given a sequence of positive terms $\{a_n\}$ with the sum of the first $n$ terms denoted as $S_n$, it satisfies the equation $2S_n = a_n^2 + a_n$ for all natural numbers $n$. Define a new sequence $\{c_n\}$ where $c_n = (-1)^n \frac{2a_n + 1}{2S_n}$. Find the sum of the first 2016 terms of the sequence $\{c_n\}$. | - \frac{2016}{2017} | 55.46875 |
32,031 | The function $f: \mathbb{R}\rightarrow \mathbb{R}$ is such that $f(x+1)=2f(x)$ for $\forall$ $x\in \mathbb{R}$ and $f(x)=x(x-1)$ for $\forall$ $x\in (0,1]$ . Find the greatest real number $m$ , for which the inequality $f(x)\geq -\frac{8}{9}$ is true for $\forall$ $x\in (-\infty , m]$ .
| 7/3 | 3.125 |
32,032 | How many distinct diagonals of a convex nonagon (9-sided polygon) can be drawn such that none of the diagonals is parallel to any side of the polygon? | 18 | 49.21875 |
32,033 | Antoine, Benoît, Claude, Didier, Étienne, and Françoise go to the cinéma together to see a movie. The six of them want to sit in a single row of six seats. But Antoine, Benoît, and Claude are mortal enemies and refuse to sit next to either of the other two. How many different arrangements are possible? | 144 | 30.46875 |
32,034 | A triangle has sides of lengths 6 cm and 8 cm that create a 45-degree angle between them. Calculate the length of the third side. | 5.67 | 0 |
32,035 | In isosceles right-angled triangle $ABC$ , $CA = CB = 1$ . $P$ is an arbitrary point on the sides of $ABC$ . Find the maximum of $PA \cdot PB \cdot PC$ . | \frac{\sqrt{2}}{4} | 43.75 |
32,036 |
Petya's bank account contains 500 dollars. The bank allows only two types of transactions: withdrawing 300 dollars or adding 198 dollars.
What is the maximum amount Petya can withdraw from the account if he does not have any other money? | 498 | 50 |
32,037 | If the graph of the function $f(x)=\sin \omega x+\sin (\omega x- \frac {\pi}{2})$ ($\omega > 0$) is symmetric about the point $\left( \frac {\pi}{8},0\right)$, and there is a zero point within $\left(- \frac {\pi}{4},0\right)$, determine the minimum value of $\omega$. | 10 | 8.59375 |
32,038 | A rectangular sheet of paper is folded so that two diagonally opposite corners come together. If the crease formed is the same length as the longer side of the sheet, what is the ratio of the longer side of the sheet to the shorter side? | \sqrt{\frac{2}{\sqrt{5} - 1}} | 0 |
32,039 | Given vectors $\overrightarrow{a}=(2\sin \omega x,2\cos \omega x)$ and $\overrightarrow{b}=(\sqrt{3}\cos\omega x,-\cos\omega x)$, where the function $f(x)=\overrightarrow{a}\cdot\overrightarrow{b}$ has a minimum positive period of $6\pi$. Find the value of the real number $\omega$. Additionally, given $α,β∈[\frac{π}{6},\frac{2π}{3}]$, where $f(3α)=-\frac{3}{13}$ and $f(3β+3π)=-\frac{11}{5}$, find $\sin(\alpha -\beta)$. | -\frac{16}{65} | 35.9375 |
32,040 | Given that the probability that a ball is tossed into bin k is 3^(-k) for k = 1,2,3,..., find the probability that the blue ball is tossed into a higher-numbered bin than the yellow ball. | \frac{7}{16} | 1.5625 |
32,041 | What value should the real number $m$ take so that the point representing the complex number $z=(m^2-8m+15)+(m^2-5m-14)i$ in the complex plane
(Ⅰ) lies in the fourth quadrant;
(Ⅱ) lies on the line $y=x$. | \frac{29}{3} | 57.8125 |
32,042 | How many ordered pairs of integers $(a, b)$ satisfy all of the following inequalities?
\[ \begin{aligned}
a^2 + b^2 &< 25 \\
a^2 + b^2 &< 8a + 4 \\
a^2 + b^2 &< 8b + 4
\end{aligned} \] | 14 | 0.78125 |
32,043 | Given \( A, B, C, D \in\{1, 2, \cdots, 6\} \), and each is distinct from the others. If the curves \( y = Ax^{2} + B \) and \( y = Cx^{2} + D \) intersect, then there are \(\quad\) different ways to choose \( A, B, C, D \) (the intersection of the curves is independent of the order of \( A, B, C, D \); for example, \( A=3, B=2, C=4, D=1 \) and \( A=4, B=1, C=3, D=2 \) are considered the same). | 90 | 3.90625 |
32,044 | Point \( M \) divides the side \( BC \) of parallelogram \( ABCD \) in the ratio \( BM : MC = 3 \). Line \( AM \) intersects the diagonal \( BD \) at point \( K \). Find the area of the quadrilateral \( CMKD \) if the area of parallelogram \( ABCD \) is 1. | 19/56 | 13.28125 |
32,045 | Given that the parabola $y=x^2-5x+2$ is symmetric about the point $(3,2)$ with $y=ax^2+bx+c$, find the value of $3a+3c+b$. | -8 | 33.59375 |
32,046 | How many non- empty subsets $S$ of $\{1,2,3,\ldots ,15\}$ have the following two properties?
$(1)$ No two consecutive integers belong to $S$.
$(2)$ If $S$ contains $k$ elements, then $S$ contains no number less than $k$. | 405 | 17.1875 |
32,047 | Rectangle ABCD has dimensions AB=CD=4 and BC=AD=8. The rectangle is rotated 90° clockwise about corner D, then rotated 90° clockwise about the corner C's new position after the first rotation. What is the length of the path traveled by point A?
A) $8\sqrt{5}\pi$
B) $4\sqrt{5}\pi$
C) $2\sqrt{2}\pi$
D) $4\sqrt{2}\pi$ | 4\sqrt{5}\pi | 37.5 |
32,048 | Given the function $g(x)=x-1$, and the function $f(x)$ satisfies $f(x+1)=-2f(x)-1$. When $x \in (0,1]$, $f(x)=x^{2}-x$. For any $x_1 \in (1,2]$ and $x_2 \in R$, determine the minimum value of $(x_1-x_2)^2+(f(x_1)-g(x_2))^2$. | \frac{49}{128} | 14.0625 |
32,049 | In the calculations shown, each letter stands for a digit. They are used to make some two-digit numbers. The two numbers on the left have a total of 79. What is the total of the four numbers on the right? | 158 | 11.71875 |
32,050 | In acute triangle $\triangle ABC$, the sides opposite angles $A$, $B$, and $C$ are denoted as $a$, $b$, and $c$ respectively. It is given that $a-4\cos C=c\cos B$.
$(1)$ Find the value of $b$;
$(2)$ If $a^{2}+b^{2}+c^{2}=2\sqrt{3}ab\sin C$, find the area of $\triangle ABC$. | 4\sqrt{3} | 40.625 |
32,051 | Point \((x,y)\) is randomly picked from the rectangular region with vertices at \((0,0), (3014,0), (3014,3015)\), and \((0,3015)\). What is the probability that \(x > 8y\)? Express your answer as a common fraction. | \frac{7535}{120600} | 0 |
32,052 | Given the sequence $\left\{a_{n}\right\}$ with its sum of the first $n$ terms $S_{n}$ satisfying $2 S_{n}-n a_{n}=n$ for $n \in \mathbf{N}^{*}$, and $a_{2}=3$:
1. Find the general term formula for the sequence $\left\{a_{n}\right\}$.
2. Let $b_{n}=\frac{1}{a_{n} \sqrt{a_{n+1}}+a_{n+1} \sqrt{a_{n}}}$ and $T_{n}$ be the sum of the first $n$ terms of the sequence $\left\{b_{n}\right\}$. Determine the smallest positive integer $n$ such that $T_{n}>\frac{9}{20}$. | 50 | 21.09375 |
32,053 | In triangle \(ABC\), side \(AB\) is 21, the bisector \(BD\) is \(8 \sqrt{7}\), and \(DC\) is 8. Find the perimeter of the triangle \(ABC\). | 60 | 7.8125 |
32,054 | We define \( a @ b = a \times (a + 1) \times \ldots \times (a + b - 1) \). Given \( x @ y @ 2 = 420 \), then \( y @ x = \) ? | 20 | 7.03125 |
32,055 | Let $p,$ $q,$ $r,$ $s$ be distinct real numbers such that the roots of $x^2 - 12px - 13q = 0$ are $r$ and $s,$ and the roots of $x^2 - 12rx - 13s = 0$ are $p$ and $q.$ Find the value of $p + q + r + s.$ | 1716 | 0.78125 |
32,056 | A bag contains 20 candies: 4 chocolate, 6 mint, and 10 butterscotch. What is the minimum number of candies that must be removed to be certain that at least two candies of each flavor have been eaten? | 18 | 63.28125 |
32,057 | Let $D$ be the circle with equation $x^2 - 4y - 4 = -y^2 + 6x + 16$. Find the center $(c,d)$ and the radius $s$ of $D$, and compute $c+d+s$. | 5 + \sqrt{33} | 96.875 |
32,058 | Michael borrows $2000$ dollars from Jane, who charges an interest of $6\%$ per month (which compounds monthly). What is the least integer number of months after which Michael will owe more than three times as much as he borrowed? | 19 | 7.03125 |
32,059 | In Papa Carlo's room, there is a clock on each wall, and they all show incorrect times: the first clock is off by 2 minutes, the second by 3 minutes, the third by 4 minutes, and the fourth by 5 minutes. One day, Papa Carlo decided to find out the exact time before leaving the house, and he saw the following times on the clocks: 14:54, 14:57, 15:02, and 15:03. Help Papa Carlo determine the exact time. | 14:58 | 11.71875 |
32,060 | Let $a,$ $b,$ and $c$ be nonzero real numbers such that $a + b + c = 3$. Simplify:
\[
\frac{1}{b^2 + c^2 - 3a^2} + \frac{1}{a^2 + c^2 - 3b^2} + \frac{1}{a^2 + b^2 - 3c^2}.
\] | -3 | 19.53125 |
32,061 | A quadrilateral connecting the midpoints of the sides of trapezoid $\mathrm{ABCD}$ is a rhombus. Find its area if the height of the trapezoid $\mathrm{BH}=5 \mathrm{c}$, the smaller base $\mathrm{BC}=6 \mathrm{~cm}$, and the angle $\mathrm{ABC}$ is $120^{\circ}$. | 15 | 1.5625 |
32,062 | A $7 \times 7$ board is either empty or contains an invisible $2 \times 2$ ship placed "by the cells." You are allowed to place detectors in some cells of the board and then activate them all at once. An activated detector signals if its cell is occupied by the ship. What is the minimum number of detectors needed to guarantee identifying whether there is a ship on the board, and if so, which cells it occupies? | 16 | 17.96875 |
32,063 | There are 5 female students and 2 male students in a class. Find the number of different distribution schemes in which they can be divided into two groups, with each group having both female and male students. | 60 | 7.8125 |
32,064 | Let $Q$ be a point chosen uniformly at random in the interior of the unit square with vertices at $(0,0), (1,0), (1,1)$, and $(0,1)$. The probability that the slope of the line determined by $Q$ and the point $\left(\frac{1}{2}, \frac{1}{4} \right)$ is greater than or equal to $1$ can be written as $\frac{p}{q}$, where $p$ and $q$ are relatively prime positive integers. Find $p+q$. | 41 | 22.65625 |
32,065 |
A circle passing through the vertex \( P \) of triangle \( PQR \) touches side \( QR \) at point \( F \) and intersects sides \( PQ \) and \( PR \) at points \( M \) and \( N \), respectively, different from vertex \( P \). Find the ratio \( QF : FR \) if it is known that the length of side \( PQ \) is 1.5 times the length of side \( PR \), and the ratio \( QM : RN = 1 : 6 \). | 1/2 | 6.25 |
32,066 | Let $q(x)$ be a quadratic polynomial such that $[q(x)]^2 - x^2$ is divisible by $(x - 2)(x + 2)(x - 5)$. Find $q(10)$. | \frac{250}{7} | 0 |
32,067 | Schools A and B are having a sports competition with three events. In each event, the winner gets 10 points and the loser gets 0 points, with no draws. The school with the higher total score after the three events wins the championship. It is known that the probabilities of school A winning in the three events are 0.5, 0.4, and 0.8, respectively, and the results of each event are independent.<br/>$(1)$ Find the probability of school A winning the championship;<br/>$(2)$ Let $X$ represent the total score of school B, find the distribution table and expectation of $X$. | 13 | 10.9375 |
32,068 | Given the equation about $x$, $(x-2)(x^2-4x+m)=0$ has three real roots.
(1) Find the range of values for $m$.
(2) If these three real roots can exactly be the lengths of the sides of a triangle, find the range of values for $m$.
(3) If the triangle formed by these three real roots is an isosceles triangle, find the value of $m$ and the area of the triangle. | \sqrt{3} | 63.28125 |
32,069 | **
Let \( P \) be a point on the parabola \( y = x^2 - 6x + 15 \), and let \( Q \) be a point on the line \( y = 2x - 7 \). Find the shortest possible distance \( PQ \).
** | \frac{6 \sqrt{5}}{5} | 6.25 |
32,070 | The regular tetrahedron, octahedron, and icosahedron have equal surface areas. How are their edges related? | 2 \sqrt{10} : \sqrt{10} : 2 | 0 |
32,071 | Let $D$ be the circle with equation $x^2 - 8x + y^2 + 10y = -14$. A translation of the coordinate system moves the center of $D$ to $(1, -2)$. Determine $a+b+r$ after the translation, where $(a,b)$ is the center and $r$ is the radius of the translated circle. | -1 + \sqrt{27} | 0 |
32,072 | Given a tetrahedron \(A B C D\), where \(B D = D C = C B = \sqrt{2}\), \(A C = \sqrt{3}\), and \(A B = A D = 1\), find the cosine of the angle between line \(B M\) and line \(A C\), where \(M\) is the midpoint of \(C D\). | \frac{\sqrt{2}}{3} | 1.5625 |
32,073 | Given the ellipse Q: $$\frac{x^{2}}{a^{2}} + y^{2} = 1 \quad (a > 1),$$ where $F_{1}$ and $F_{2}$ are its left and right foci, respectively. A circle with the line segment $F_{1}F_{2}$ as its diameter intersects the ellipse Q at exactly two points.
(1) Find the equation of ellipse Q;
(2) Suppose a line $l$ passing through point $F_{1}$ and not perpendicular to the coordinate axes intersects the ellipse at points A and B. The perpendicular bisector of segment AB intersects the x-axis at point P. The range of the x-coordinate of point P is $[-\frac{1}{4}, 0)$. Find the minimum value of $|AB|$. | \frac{3\sqrt{2}}{2} | 15.625 |
32,074 | Consider a terminal with fifteen gates arranged in a straight line with exactly $90$ feet between adjacent gates. A passenger's departure gate is assigned at random. Later, the gate is changed to another randomly chosen gate. Calculate the probability that the passenger walks $360$ feet or less to the new gate. Express the probability as a simplified fraction $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers, and find $m+n$. | 31 | 2.34375 |
32,075 | In the geometric sequence ${a_n}$ where $q=2$, if the sum of the series $a_2 + a_5 + \dots + a_{98} = 22$, calculate the sum of the first 99 terms of the sequence $S_{99}$. | 77 | 51.5625 |
32,076 | Jack Sparrow needed to distribute 150 piastres into 10 purses. After putting some amount of piastres in the first purse, he placed more in each subsequent purse than in the previous one. As a result, the number of piastres in the first purse was not less than half the number of piastres in the last purse. How many piastres are in the 6th purse? | 15 | 15.625 |
32,077 | The sequence of natural numbers $1, 5, 6, 25, 26, 30, 31,...$ is made up of powers of $5$ with natural exponents or sums of powers of $5$ with different natural exponents, written in ascending order. Determine the term of the string written in position $167$ . | 81281 | 29.6875 |
32,078 | Point \((x,y)\) is randomly picked from the rectangular region with vertices at \((0,0), (3000,0), (3000,3001),\) and \((0,3001)\). What is the probability that \(x > 3y\)? Express your answer as a common fraction. | \frac{1500}{9003} | 0 |
32,079 | In triangle $XYZ$, the medians $\overline{XT}$ and $\overline{YS}$ are perpendicular. If $XT = 15$ and $YS = 20$, find the length of side $XZ$. | \frac{50}{3} | 10.9375 |
32,080 | Given a function $f(x)$ that satisfies: (1) For any positive real number $x$, it always holds that $f(2x) = 2f(x)$; (2) When $x \in (1, 2)$, $f(x) = 2 - x$. If $f(a) = f(2020)$, determine the smallest positive real number $a$. | 36 | 14.84375 |
32,081 | Find the value of the expression $$ f\left( \frac{1}{2000} \right)+f\left( \frac{2}{2000} \right)+...+ f\left( \frac{1999}{2000} \right)+f\left( \frac{2000}{2000} \right)+f\left( \frac{2000}{1999} \right)+...+f\left( \frac{2000}{1} \right) $$ assuming $f(x) =\frac{x^2}{1 + x^2}$ . | 1999.5 | 1.5625 |
32,082 | A train has five carriages, each containing at least one passenger. Two passengers are said to be 'neighbours' if either they are in the same carriage or they are in adjacent carriages. Each passenger has exactly five or exactly ten neighbours. How many passengers are there on the train? | 17 | 4.6875 |
32,083 | For the set $\{1, 2, 3, \ldots, 8\}$ and each of its non-empty subsets, a unique alternating sum is defined as follows: arrange the numbers in the subset in decreasing order and then, beginning with the largest, alternately add and subtract successive numbers. Calculate the sum of all such alternating sums for $n=8$. | 1024 | 75.78125 |
32,084 | Given point P(2, 1) is on the parabola $C_1: x^2 = 2py$ ($p > 0$), and the line $l$ passes through point Q(0, 2) and intersects the parabola $C_1$ at points A and B.
(1) Find the equation of the parabola $C_1$ and the equation for the trajectory $C_2$ of the midpoint M of chord AB;
(2) If lines $l_1$ and $l_2$ are tangent to $C_1$ and $C_2$ respectively, and $l_1$ is parallel to $l_2$, find the shortest distance from $l_1$ to $l_2$. | \sqrt{3} | 13.28125 |
32,085 | The sum of the first $n$ terms of the sequence $\{a_n\}$ is $S_n=n^2+n+1$, and $b_n=(-1)^n(a_n-2)$ $(n\in\mathbb{N}^*)$, then the sum of the first $50$ terms of the sequence $\{b_n\}$ is $\_\_\_\_\_\_\_$. | 49 | 38.28125 |
32,086 | The Sequence $\{a_{n}\}_{n \geqslant 0}$ is defined by $a_{0}=1, a_{1}=-4$ and $a_{n+2}=-4a_{n+1}-7a_{n}$ , for $n \geqslant 0$ . Find the number of positive integer divisors of $a^2_{50}-a_{49}a_{51}$ . | 51 | 10.9375 |
32,087 | To determine the minimum time required for the concentration of the drug in the air to drop below 0.25 milligrams per cubic meter, solve the equation y = 0.25 for t using the function y=\begin{cases} 10t & (0 \leqslant t \leqslant 0.1), \\ {\left( \frac{1}{16} \right)}^{t- \frac{1}{10}} & (t > 0.1) \end{cases}. | 0.6 | 14.0625 |
32,088 | Given the function $f(x)=\frac{cos2x+a}{sinx}$, if $|f(x)|\leqslant 3$ holds for any $x\in \left(0,\pi \right)$, then the set of possible values for $a$ is ______. | \{-1\} | 0 |
32,089 | How many different rectangles with sides parallel to the grid can be formed by connecting four of the dots in a $5\times 5$ square array of dots? | 225 | 91.40625 |
32,090 | Given $f(x)=9^{x}-2×3^{x}+4$, where $x\in\[-1,2\]$:
1. Let $t=3^{x}$, with $x\in\[-1,2\}$, find the maximum and minimum values of $t$.
2. Find the maximum and minimum values of $f(x)$. | 67 | 22.65625 |
32,091 | Find the period of the repetend of the fraction $\frac{39}{1428}$ by using *binary* numbers, i.e. its binary decimal representation.
(Note: When a proper fraction is expressed as a decimal number (of any base), either the decimal number terminates after finite steps, or it is of the form $0.b_1b_2\cdots b_sa_1a_2\cdots a_ka_1a_2\cdots a_ka_1a_2 \cdots a_k \cdots$ . Here the repeated sequence $a_1a_2\cdots a_k$ is called the *repetend* of the fraction, and the smallest length of the repetend, $k$ , is called the *period* of the decimal number.) | 24 | 3.125 |
32,092 | The diagram shows a solid with six triangular faces and five vertices. Andrew wants to write an integer at each of the vertices so that the sum of the numbers at the three vertices of each face is the same. He has already written the numbers 1 and 5 as shown. What is the sum of the other three numbers he will write? | 11 | 10.15625 |
32,093 | Construct a circle with center \( S \) and radius \( 3 \text{ cm} \). Construct two mutually perpendicular diameters \( AC \) and \( BD \) of this circle. Construct isosceles triangles \( ABK, BCL, CDM, DAN \) such that:
- The base of each triangle is a side of the quadrilateral \( ABCD \).
- The base of each triangle is equal to the height on this side.
- No triangle overlaps the quadrilateral \( ABCD \).
From the given data, calculate the area of the polygon \( AKBLCDMN \). | 108 | 9.375 |
32,094 | Alexio has 120 cards numbered from 1 to 120, inclusive, and places them in a box. He then randomly picks a card. What is the probability that the number on the card is a multiple of 2, 4, or 6? Express your answer as a common fraction. | \frac{1}{2} | 30.46875 |
32,095 | Given a right triangle \(ABC\), point \(D\) is located on the extension of hypotenuse \(BC\) such that line \(AD\) is tangent to the circumcircle \(\omega\) of triangle \(ABC\). Line \(AC\) intersects the circumcircle of triangle \(ABD\) at point \(E\). It turns out that the angle bisector of \(\angle ADE\) is tangent to circle \(\omega\). What is the ratio in which point \(C\) divides segment \(AE\)? | 1:2 | 25.78125 |
32,096 | If $\mathbf{a}$, $\mathbf{b}$, $\mathbf{c}$, and $\mathbf{d}$ are unit vectors, then find the largest possible value of
\[
\|\mathbf{a} - \mathbf{b}\|^2 + \|\mathbf{a} - \mathbf{c}\|^2 + \|\mathbf{a} - \mathbf{d}\|^2 + \|\mathbf{b} - \mathbf{c}\|^2 + \|\mathbf{b} - \mathbf{d}\|^2 + \|\mathbf{c} - \mathbf{d}\|^2.
\] | 16 | 21.875 |
32,097 | Rectangle $EFGH$ has area $4032$. An ellipse with area $4032\pi$ passes through points $E$ and $G$ and has foci at $F$ and $H$. Determine the perimeter of the rectangle $EFGH$. | 8\sqrt{2016} | 0 |
32,098 | How many whole numbers between 1 and 2000 do not contain the digit 7? | 1457 | 0 |
32,099 | A certain orange orchard has a total of 120 acres, consisting of both flat and hilly land. To estimate the average yield per acre, a stratified sampling method is used to survey a total of 10 acres. If the number of hilly acres sampled is 2 times plus 1 acre more than the flat acres sampled, then the number of acres of flat and hilly land in this orange orchard are respectively \_\_\_\_\_\_\_\_ and \_\_\_\_\_\_\_\_. | 84 | 68.75 |
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